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Thread: Recursive and Special Sequence word problem

  1. #1

    Recursive and Special Sequence word problem

    Join two 1-unit by 1 unit squares to form a rectangle. Next, draw a larger square along a long side of the rectangle. Continue this process.

    a) Write the sequence of the lengths of the sides of the squares you added at each step. Befin the sequence with two original squares.

    b) Write a recursive formula for the sequence of lengths added.

    c) Identify the sequence in b.

    I dont know how to do any of it. step 1 has 2 squarws step 2 has 6 squares and step 3 has 15 square.
    Karlie McQueen

  2. #2
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    Re: Recursive and Special Sequence word problem

    Join two 1-unit by 1 unit squares to form a rectangle. Next, draw a larger square along a long side of the rectangle. Continue this process.

    a) Write the sequence of the lengths of the sides of the squares you added at each step. Befin the sequence with two original squares.

    b) Write a recursive formula for the sequence of lengths added.

    c) Identify the sequence in b.

    I dont know how to do any of it. step 1 has 2 squarws step 2 has 6 squares and step 3 has 15 square.

    Not quite sure as to what you mean.

    1--Two 1 x 1 squares
    2--Attach a 2 x 2 square to the 2 side of the rectangle making a 2 x 4 rectangle.
    3--Attach a matching 4 x 4 square to the 4 side making a 4 x 8 rectangle.
    4-- Attach a matching 8 x 8 square to the 8 side making an 8 x 16 rectangle.
    5--...

    Squares added:
    1--1
    2--8
    3--32
    4--128
    5--...

    Lengths of added square sides:
    1--2
    2--4
    3--8
    4--16
    5--...
    ????????????????????

    Could you clarify?
    TchrWill

    No matter how insignificant it might appear, learn something new every day.

  3. #3
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    Re: Recursive and Special Sequence word problem

    Quote Originally Posted by karliekay
    Join two 1-unit by 1 unit squares to form a rectangle. Next, draw a larger square along a long side of the rectangle. Continue this process.

    a) Write the sequence of the lengths of the sides of the squares you added at each step. Befin the sequence with two original squares.

    b) Write a recursive formula for the sequence of lengths added.

    c) Identify the sequence in b.

    I dont know how to do any of it. step 1 has 2 squarws step 2 has 6 squares and step 3 has 15 square.
    The number of squares increases by one each time, but I'm not using the number of squares for the pattern.


    a) Sequence of the lengths of the sides:

    1, 1, 2, 3, 5, 8, ...



    b) [tex]\ \ a_{n + 2} \ =\ a_{n + 1} \ + \ a_{n}, \ where \ a_0 \ = \ 1 \ and \ a_1 \ =\ 1.[/tex]



    c) Fibonacci sequence



    Did I show too much?

  4. #4
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    Re: Recursive and Special Sequence word problem

    lookagain found the right path. Nice going.

    Just as an afterthought, the nth term of the Fibonacci Sequence is given by
    Fn = [((1 + sqrt5)/2)^n - ((1 -sqrt5)/2)^n]/sqrt(5)

    If you are interested in some interesting tidbits about the Fibonacci Sequence, read on.

    Closer examination of the series of numbers will lead you to some surprisingly interesting and unique relationships between the Fibonacci numbers Fn = F(n-1) + F(n-2).

    As n increases, the ratio of F(n+1)/Fn approaches the Golden Ratio, (1 + sqrt5)/2 = 1.618....

    The sum of the squares of two adjacent Fibonacci numbers is equal to a higher Fibonacci number according to Fn^2 + F(n+1)^2 = F(2n+1). For instance, the 4thFn^2 + the 5thFn^2 = the F(2(4) + 1) = 9th Fn or 3^2 + 5^2 = 34, the 9th Fn.

    The product of two alternating Fibonacci numbers minus the square of the one in between is equal to +/- one as expressed by F(n-1)F(N+1) - Fn^2 = (-1)^n. For instance, 8x21 - 13^2 = 168 - 169 = (-1)^7 = -1.

    The sum of the cubes of two adjacent Fibonacci numbers minus the cube of the preceding one is equal to a higher Fibonacci number as expressed by Fn^3 + F(n+1)^3 = F3n. For instance, F4^3 + F5^3 - F3^3 = 3^3 + 5^3 - 2^3 = 27 + 125 - 8 = 144 = F12 = 144.

    The sum of the squares of a series of Fibonacci numbers starting with F1 and ending with Fn is equal to the product of Fn and F(n+1) as expressed by F1^2 + F2^2 + F3^2 + ......Fn^2 = FnF(n+1). For instance, 1^2 + 1^2 + 2^2 + 3^2 + 5^2 = 1 + 1 + 4 + 9 + 25 = 40 = 5x8.

    The sum of a series of Fibonacci numbers starting with F1 and ending with Fn is equal to a higher Fibonacci number as expressed by F1 + F2 + F3 + .......Fn = F(n+2) - 1. For instance, 1 + 1 + 2 + 3 + 5 + 8 = 20 = 21 - 1.

    The sum of any 10 consecutive Fibonacci numbers is 11 times the 7th term of the 10 numbers.
    For instance, the sum of the 4th through 13th numbers, 3,5,8,13,21,34,55,89,144,233, is 11x55 = 605.

    The sum of any number of consecutive Fibonacci numbers is given by S[Fn1-->Fn2] = F(n2+2) - F(n1+1).
    For instance, the sum of the 5th through 10th numbers, 5,8,13,21,34,55, is 144 - 8 = 136.

    The sum of the first n even terms of Fibonacci numbers is given by F2 + F4 + F6 + F8 + ....F2n = F(2n+1) - 1.
    For instance, the sum of the first 6 even terms is therefore F(2(6) + 1) - 1 = F(13) - 1 = 233 - 1 = 232.

    The sum of the first n odd terms of Fibonacci numbers is given by F1 + F3 + F5 + ....F(2n-1) = F(2n)
    For instance, the sum of the first 6 odd terms is therefore F(2n) = F(12) = 144.

    Fn^2 + F(n+1)^2 = F(2n+1)

    F(n+1)F(n-1) - Fn^2 = (-1)^n

    Some other relationships of Fibonacci numbers are Fn^2 + F(n+1)^2 = F(2n+1) and F(n+1)F(n-1) - Fn^2 = (-1)^n.

    The ratio of the 2nth Fibonacci number divided by the nth Fibonacci number is always an integer or F2n/Fn = K. For instance, F10/F5 = 55/5 = 11.

    The sum of any 4n consecutive Fibonacci numbers is evenly divisible by F2n.
    Example: The sum of 4(2) = 8 Fibonacci numbers is divisible by F2(2) = F4 = 3.
    1+2+3+5+8+13+21+34 = 87/3 = 29.
    2+3+5+8+13+21+34+55 = 141/3 = 47.
    3+5+8+13+21+34+55+89 = 228/3 = 76

    As any number may be represented by combinations of powers of 2, so may any number be represented by combinations of the Fibonacci numbers.
    1 = 1, 2 = 2, or 1+1, 3 = 3 or 1+2, 4 = 3+1, 5 = 5 or 3+2, 6 = 5+1 or 3+2+1, 7 = 5+2, 8 = 5+3, 9=5+3+1, 10 = 5+3+2, 50 = 34+13+3, 100 = 89+8+3 or 55+34+8+3, and so on.

    The greatest common divisor of any two Fibonacci numbers is, itself, a Fibonacci number. Even more surprising is the fact that the g.c.d.(Fa, Fb) = c = F[g.c.d.(a,b)]. What this means is that the g.c.d. of the "a"th and "b"th Fibonacci numbers is the "c"th Fibonacci number where c = the g.c.d. of a and b.

    For any two consecutive Fibonacci numbers, a and b, a^2 - ab - b^2 = +1 or -1.

    Given the four consecutive Fibonacci numbers a, b, c and d, (ad), 2(bc) and (cd - ab) form a Pythagorean Triple such that (ad)^2 + [2(bc)]^2 = (cd - ab)^2.

    Given two positive integers "m" and "n" that are relatively prime, the Fibonacci numbers F(m) and F(n) are also relatively prime.
    TchrWill

    No matter how insignificant it might appear, learn something new every day.

  5. #5
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    Re: Recursive and Special Sequence word problem

    Hello, karliekay!

    Join two 1-by-1 squares to form a rectangle.
    . . Next, draw a larger square along a long side of the rectangle.
    . . Continue this process.

    a) Write the sequence of the lengths of the sides of the squares you added at each step.
    . . Begin the sequence with two original squares.

    We start with one square.

    Code:
          *---*
        1 |   |
          *---*
            1


    Add a square:

    Code:
          *---*
          |   |
          *---*
          |:::| 1
          *---*
            1


    Add a square:

    Code:
          *---*-------*
        1 |   |:::::::|
          *---*:::::::| 2
        1 |   |:::::::|
          *---*-------*
                  2


    Add a square.

    Code:
          *---*-------*
        1 |   |       |
          *---*       | 2
        1 |   |       |
          *---*-------*
          |:::::::::::|
          |:::::::::::|
        3 |:::::::::::| 3
          |:::::::::::|
          *-----------*
                3


    Add a square.

    Code:
          *---*-------*-----------------*
        1 |   |       |:::::::::::::::::|
          *---*       |:::::::::::::::::|
        1 |   |       |:::::::::::::::::|
          *---*-------*:::::::::::::::::|
          |           |:::::::::::::::::| 5
          |           |:::::::::::::::::|
        3 |           |:::::::::::::::::|
          |           |:::::::::::::::::|
          *-----------*-----------------*
                3              5

    [tex]\text{It appears that the sequence is:}\;1,\,1,\,2,\,3,\,5,\,8,\,13\:\hdots[/tex]




    b) Write a recursive formula for the sequence of lengths added.

    [tex]\text{Each term is the sum of the preceding two terms.}[/tex]

    . . [tex]F_n \;=\;F_{n-1} + F_{n-2}[/tex]




    c) Identify the sequence in b.

    [tex]\text{This is the Fibonacci sequence.}[/tex]

    I'm the other of the two guys who "do" homework.

  6. #6
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    Re: Recursive and Special Sequence word problem

    That problem was my first exposure to Fibonacci - through old Scientific American - back in early 60's.
    “... mathematics is only the art of saying the same thing in different words” - B. Russell

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